Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-14T03:44:22.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenstructure of the infinitely-many-neutral-alleles diffusion model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

S. N. Ethier
S. N. Ethier*
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. Research supported in part by NSF grant DMS-8902991.
Get access Rights & Permissions [Opens in a new window]

Abstract

The complete set of eigenvalues is found for the (unlabeled) infinitely-many-neutral-alleles diffusion model. The transition density for the process, originally derived by Griffiths, is rederived as an eigenfunction expansion.

Keywords

POPULATION GENETICSINFINITE-DIMENSIONAL DIFFUSION PROCESSPOISSON-DIRICHLET DISTRIBUTIONEWENS' SAMPLING FORMULASYMMETRIC OPERATOREIGENFUNCTION EXPANSIONTRANSITION DENSITY

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 92D15: Problems related to evolution
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 487 - 498
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ethier, S. N. and Kurtz, T. G. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.Google Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Ewens, W. J. and Kirby, K. (1975) The eigenvalues of the neutral alleles process. Theoret. Popn. Biol. 7, 212–220.CrossRefGoogle ScholarPubMed
Griffiths, R. C. (1979a) A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325.CrossRefGoogle Scholar
Griffiths, R. C. (1979b) Exact sampling distribution from the infinite neutral alleles model. Adv. Appl. Prob. 11, 326–354.Google Scholar
Karlin, S. and Avni, H. (1975) Derivation of the eigenvalues of the configuration process induced by a labeled direct product branching process. Theoret. Popn Biol. 7, 221–228.Google Scholar
Kingman, J. F. C. (1975) Random discrete distributions. J. R. Statist. Soc. B37, 1–22.Google Scholar
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn Biol. 11, 274–283.CrossRefGoogle ScholarPubMed
Schmuland, B. (1991) A result on the infinitely many neutral alleles diffusion model. J. Appl. Prob. 28, 253–267.CrossRefGoogle Scholar
Watterson, G. A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.CrossRefGoogle Scholar